For one of the easier bonds, most facilitated by other bonds (such as 2 × 5 = 10, or 10 - 2 = 8, or the double bond 7 = two 3s and 1 remainder) in the case of the median or average pupil, twelve practices in the week of first learning, supported by twenty-five practices during the two months following, and maintained by thirty practices well spread over the later periods should be enough. For the more gifted pupils lesser amounts down to six, twelve, and fifteen may suffice. For the less gifted pupils more may be required up to thirty, fifty, and a hundred. It is to be doubted, however, whether pupils requiring nearly two hundred repetitions of each of these easy bonds should be taught arithmetic beyond a few matters of practical necessity. For bonds of ordinary difficulty, with average facilitation from other bonds (such as 11 - 3, 4 × 7, or 48 ÷ 8 = 6) in the case of the median or average pupil, we may estimate twenty practices in the week of first learning, supported by thirty, and maintained by fifty practices well spread over the later periods. Gifted pupils may gain and keep mastery with twelve, fifteen, and twenty practices respectively. Pupils dull at arithmetic may need up to twenty, sixty, and two hundred. Here, again, it is to be doubted whether a pupil for whom arithmetical facts, well taught and made interesting, are so hard to acquire as this, should learn many of them. For bonds of greater difficulty, less facilitated by other bonds (such as 17 - 9, 8 × 7, or 12-1/2% of = 1/8 of), the practice may be from ten to a hundred percent more than the above. UNDERLEARNING AND OVERLEARNING If we accept the above provisional estimates as reasonable, we may consider the harm done by giving less and by giving more than these reasonable amounts. Giving less is indefensible. The pupil's time is wasted in excessive checking to find his errors. He is in danger of being practiced in error. His attention is diverted from the learning of new facts and processes by the necessity of thinking out these supposedly mastered facts. All new bonds are harder to learn than they should be because the bonds which should facilitate them are not strong enough to do so. Giving more does harm to some extent by using up time that could be spent better for other purposes, and (though not necessarily) by detracting from the pupil's interest in arithmetic. In certain cases, however, such excess practice and overlearning are actually desirable. Three cases are of special importance. The first is the case of a bond operating under a changed mental set or adjustment. A pupil may know 7 × 8 adequately as a thing by itself, but need more practice to operate it in 285 7 ---